|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| The '''Malliavin calculus''', named after [[Paul Malliavin]], extends the [[calculus of variations]] from functions to [[stochastic processes]]. The Malliavin calculus is also called the '''[[stochastic calculus]] of variations'''. In particular, it allows the computation of [[derivative]]s of [[random variable]]s.
| | My name: Alisia Combs<br>Age: 24<br>Country: United Kingdom<br>City: Aultguish Inn <br>Post code: Iv23 7sg<br>Address: 76 Leicester Road<br><br>Here is my weblog :: [http://www.77groups.com/blogs/viewstory/7978 service dog training schools] |
| | |
| Malliavin ideas led to a proof that [[Hörmander's condition]] implies the existence and smoothness of a [[probability density function|density]] for the solution of a [[stochastic differential equation]]; [[Lars Hörmander|Hörmander]]'s original proof was based on the theory of [[partial differential equation]]s. The calculus has been applied to [[stochastic partial differential equation]]s as well.
| |
| | |
| The calculus allows [[integration by parts]] with [[random variable]]s; this operation is used in [[mathematical finance]] to compute the sensitivities of [[derivative (finance)|financial derivative]]s. The calculus has applications for example in [[stochastic filtering]].
| |
| | |
| ==Overview and history==
| |
| Paul Malliavin's [[stochastic calculus]] of variations extends the [[calculus of variations]] from functions to [[stochastic processes]]. In particular, it allows the computation of [[derivative]]s of [[random variable]]s.
| |
| | |
| Malliavin invented his calculus to provide a stochastic proof that [[Hörmander's condition]] implies the existence of a [[probability density function|density]] for the solution of a [[stochastic differential equation]]; [[Lars Hörmander|Hörmander]]'s original proof was based on the theory of [[partial differential equation]]s. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to [[stochastic partial differential equation]]s.
| |
| | |
| == Invariance principle ==
| |
| The usual invariance principle for [[Lebesgue integration]] over the whole real line is that, for any real number ε and integrable function ''f'', the
| |
| following holds
| |
| | |
| :<math> \int_{-\infty}^\infty f(x)\, d \lambda(x) = \int_{-\infty}^\infty f(x+\varepsilon)\, d \lambda(x) .</math> | |
| | |
| This can be used to derive the [[integration by parts]] formula since, setting ''f'' = ''gh'' and differentiating with respect to ε on both sides, it implies
| |
| | |
| :<math> \int_{-\infty}^\infty f' \,d \lambda = \int_{-\infty}^\infty (gh)' \,d \lambda = \int_{-\infty}^\infty g h'\, d \lambda + | |
| \int_{-\infty}^\infty g' h\, d \lambda.</math>
| |
| | |
| A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let <math>h_s</math> be a square-integrable [[predictable process]] and set
| |
| | |
| :<math> \varphi(t) = \int_0^t h_s\, d s .</math> | |
| | |
| If <math>X</math> is a [[Wiener process]], the [[Girsanov theorem]] then yields the following analogue of the invariance principle:
| |
| | |
| :<math> E(F(X + \varepsilon\varphi))= E \left [F(X) \exp \left ( \varepsilon\int_0^1 h_s\, d X_s - | |
| \frac{1}{2}\varepsilon^2 \int_0^1 h_s^2\, ds \right ) \right ].</math>
| |
| | |
| Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:
| |
| | |
| :<math>E(\langle DF(X), \varphi\rangle) = E\Bigl[ F(X) \int_0^1 h_s\, dX_s\Bigr]. | |
| </math> | |
| | |
| Here, the left-hand side is the [[Malliavin derivative]] of the random variable <math>F</math> in the direction <math>\varphi</math> and the integral appearing on the right hand side should be interpreted as an [[Itô integral]]. This expression also remains true (by definition) if <math>h</math> is not adapted, provided that the right hand side is interpreted as a [[Skorokhod integral]].{{Citation needed|date=August 2011}} | |
| | |
| == Clark-Ocone formula ==
| |
| {{Main|Clark–Ocone theorem}}
| |
| | |
| One of the most useful results from Malliavin calculus is the [[Clark-Ocone theorem]], which allows the process in the [[martingale representation theorem]] to be identified explicitly. A simplified version of this theorem is as follows:
| |
| | |
| For <math>F: C[0,1] \to \R</math> satisfying <math> E(F(X)^2) < \infty</math> which is Lipschitz and such that ''F'' has a strong derivative kernel, in the sense that
| |
| for <math>\varphi</math> in ''C''[0,1]
| |
| | |
| :<math> \lim_{\varepsilon \to 0} (1/\varepsilon)(F(X+\varepsilon \varphi) - F(X) ) = \int_0^1 F'(X,dt) \varphi(t)\ \mathrm{a.e.}\ X</math>
| |
| | |
| then
| |
| | |
| :<math>F(X) = E(F(X)) + \int_0^1 H_t \,d X_t ,</math>
| |
| | |
| where ''H'' is the previsible projection of ''F''<nowiki>'</nowiki>(''x'', (''t'',1]) which may be viewed as the derivative of the function ''F'' with respect to a suitable parallel shift of the process ''X'' over the portion (''t'',1] of its domain.
| |
| | |
| This may be more concisely expressed by
| |
| | |
| :<math>F(X) = E(F(X))+\int_0^1 E (D_t F | \mathcal{F}_t ) \, d X_t .</math>
| |
| | |
| Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals ''F'' by replacing the derivative kernel used above by the "[[Malliavin derivative]]" denoted <math>D_t</math> in the above statement of the result. {{Citation needed|date=August 2011}}
| |
| | |
| == Skorokhod integral ==
| |
| {{Main|Skorokhod integral}}
| |
| The [[Skorokhod integral]] operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative thus for u in the domain of the operator which is a subset of <math>L^2([0,\infty) \times \Omega)</math>,
| |
| for '''F''' in the domain of the Malliavin derivative, we require
| |
| | |
| : <math> E (\langle DF, u \rangle ) = E (F \delta (u) ),</math>
| |
| | |
| where the inner product is that on <math>L^2[0,\infty)</math> viz
| |
| | |
| : <math> \langle f, g \rangle = \int_0^\infty f(s) g(s) \, ds.</math>
| |
| | |
| The existence of this adjoint follows from the [[Riesz representation theorem]] for linear operators on [[Hilbert spaces]].
| |
| | |
| It can be shown that if ''u'' is adapted then
| |
| | |
| : <math> \delta(u) = \int_0^\infty u_t\, d W_t ,</math>
| |
| | |
| where the integral is to be understood in the Itô sense. Thus this provides a method of extending the Itô integral to non adapted integrands.
| |
| | |
| ==Applications==
| |
| The calculus allows [[integration by parts]] with [[random variable]]s; this operation is used in [[mathematical finance]] to compute the sensitivities of [[derivative (finance)|financial derivative]]s. The calculus has applications for example in [[stochastic control|stochastic filtering]].
| |
| | |
| {{No footnotes|date=June 2011}}
| |
| | |
| == References ==
| |
| * Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", ''Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto'' 1982, pp 271–306
| |
| * Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", ''J. Faculty Sci. Uni. Tokyo Sect. 1A Math.'', 32 pp 1–76
| |
| * Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", ''J. Faculty Sci. Univ. Tokyo Sect. 1A Math.'', 34 pp 391–442
| |
| * Malliavin, Paul and Thalmaier, Anton. ''Stochastic Calculus of Variations in Mathematical Finance'', Springer 2005, ISBN 3-540-43431-3
| |
| * {{cite book
| |
| | last = Nualart
| |
| | first = David
| |
| | title = The Malliavin calculus and related topics
| |
| | edition = Second edition
| |
| |publisher = Springer-Verlag
| |
| | year = 2006
| |
| | isbn = 978-3-540-28328-7
| |
| }}
| |
| * Bell, Denis. (2007) ''The Malliavin Calculus'', Dover. ISBN 0-486-44994-7
| |
| * Schiller, Alex (2009) [http://www.alexschiller.com/media/Thesis.pdf ''Malliavin Calculus for Monte Carlo Simulation with Financial Applications'']. Thesis, Department of Mathematics, Princeton University
| |
| * [[Bernt Øksendal|Øksendal, Bernt K.]].(1997) [http://www.quantcode.com/modules/wflinks/visit.php?cid=11&lid=4 ''An Introduction To Malliavin Calculus With Applications To Economics'']. Lecture Notes, Dept. of Mathematics, University of Oslo (Zip file containing Thesis and addendum)
| |
| *Di Nunno, Giulia, Øksendal, Bernt, Proske, Frank (2009) "Malliavin Calculus for Lévy Processes with Applications to Finance", Universitext, Springer. ISBN 978-3-540-78571-2
| |
| | |
| == External links ==
| |
| | |
| * {{cite web
| |
| |url = http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf
| |
| |title = An Introduction to Malliavin Calculus
| |
| |accessdate = 2007-07-23
| |
| |last = Friz
| |
| |first = Peter K.
| |
| |date = 2005-04-10
| |
| |format = PDF
| |
| |archiveurl = http://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf |archivedate = 2007-04-17}} Lecture Notes, 43 pages
| |
| | |
| [[Category:Stochastic calculus]]
| |
| [[Category:Integral calculus]]
| |
| [[Category:Mathematical finance]]
| |
| [[Category:Calculus of variations]]
| |